Leonhard Euler, the magnificent 18th century Swiss mathematician, was unable to prove Fermat’s Last Theorem, as we mentioned in the previous episode. The story goes that he sent a friend to Pierre Fermat’s house (who had been dead for almost eighty years by that time) in the hope of finding, in some crumbling document, a glimpse of the “marvellous proof” so cruelly withheld from the world’s scrutiny. But no, nothing, gar nichts…
Leonhard Euler is the mathematical equivalent of Johann Sebastian Bach, being highly gifted, hard working and excessively fertile – speaking both in literal and figurative terms. He fathered about thirteen children, and he is said to have been able to perform difficult calculations while dandling one of them on his knee. Euler is the genius who solved the so-called Basel Problem (the Swiss town of Basel being a centre of mathematics at the time, though Euler himself spent most of his working life in St Petersburg and Berlin). The Basel Problem can be described as – what is the solution of the next sum:
Well, what would you think? Does it run off to infinity? The number of terms is certainly endless. Does this mean that the sum is also infinite? Or is it that this sum, asymptotically, approaches some fixed number? And if so, what number?
Apart from finding that the second answer was indeed correct, Euler also provided his audience with the proper term – which here I will write with a capital sigma for the sum:
This is a really perplexing outcome! Having the ratio between the radius and the circumference of a circle returning in a sum like this! It is true beauty, call it a stroke of God’s brush…
Leonard Euler – excessively fertile in all respects
If Euler was unable to prove Fermat’s Last Theorem, he nevertheless managed to generalize it. A software developer would use the term “making it more generic.” Before I really go into this, I want to mention the fact that, in the present day, Fermat has been fully generalized. The question then becomes: – how many terms do you need on the left side, and how many terms do you need on the right side in order to find solutions for whole numbers having n as an exponent? In order to describe these equations, a simple threefold notation is used (n, m, k) – in which n is the power, and m and k are respectively the number of terms at the left and the right side.
Are you still with me?
So the (3,2,2)-problem can be rephrased as – are there any solutions for whole numbers for:
a3 + b3 = z3 + y3
Well, yes. In the next example (one suffices of course) the two terms on both sides sum up to 1729:
123 + 13 = 103 + 93
What about (3,3,1)? Also here the answer is positive. For example:
33 + 43 + 53 = 63
And what about higher powers? It is not difficult to see that you can guarantee a solution simply by making the number of terms on one the sides large enough. Starting from a found solution you then can start reducing the number of terms on one of the sides, examining where things start to get troublesome. Where lies the frontier between the solution and the no-solution area? What is the shape of this landscape? Here, the computer comes in. Quite a lot of energy has been spent on charting this domain – for instance, take a look at http://euler.free.fr/index.htm.
Now back to Euler. It was his conjecture that, assuming the number of terms on one of the sides to be equal to one, the number of terms on the other side must equal the power governing the equation. To take the simplest, that is, the 4th power as an example – Euler’s Conjecture states that there are no solutions for whole numbers for:
a4 + b4 + c4 = z4
Using the notation we mentioned above – equations of type (n, k, 1) will only allow solutions if k >= n.
Well, what you think? Is this conjecture true? Aesthetically, I would say yes. If the universe is ordered in such a way that π can crop up in a sum of reciprocal squares, than also this conjecture made by Euler – for he made quite a number – is likely to be true. Yet, to the consternation of a great number of mathematicians, and only in a world in which computers exist, it turned out that this conjecture could be disproved by numerical examples.